The groundbreaking technology also means for the first time tissue transplants can be carried out without the need for anti-rejection drugs.
Five months on the patient, 30-year-old mother-of-two Claudia Castillo, is in perfect health, The Lancet reports.
She needed the transplant to save a lung after contracting tuberculosis.
The Colombian woman's airways had been damaged by the disease.
Scientists from Bristol helped grow the cells for the transplant and the European team believes such tailor-made organs could become the norm.
To make the new airway, the doctors took a donor windpipe, or trachea, from a patient who had recently died.
Then they used strong chemicals and enzymes to wash away all of the cells from the donor trachea, leaving only a tissue scaffold made of the fibrous protein collagen.
This gave them a structure to repopulate with cells from Ms Castillo herself, which could then be used in an operation to repair her damaged left bronchus - a branch of the windpipe.
By using Ms Castillo's own cells the doctors were able to trick her body into thinking the donated trachea was part of it, thus avoiding rejection.
Two types of cell were taken from Ms Castillo: cells lining her windpipe, and adult stem cells - very immature cells from the bone marrow - which could be encouraged to grow into the cells that normally surround the windpipe.
I was very much afraid. Before this, we had been doing this work only in pigs
After four days of growth in the lab in a special rotating bioreactor, the newly-coated donor windpipe was ready to be transplanted into Ms Castillo.
Her surgeon, Professor Paolo Macchiarini of the Hospital Clínic of Barcelona, Spain, carried out the operation in June
He said: "I was very much afraid. Before this, we had been doing this work only on pigs.
"But as soon as the donor trachea came out of the bioreactor it was a very positive surprise."
He said it looked and behaved identically to a normal human donor trachea.
The trachea graft ready for transplantation
The operation was a great success and just four days after transplantation the hybrid windpipe was almost indistinguishable from adjacent normal airways.
After a month, a biopsy of the site proved that the transplant had developed its own blood supply.
And with no signs of rejection four months on, Professor Macchiarini says the future chance of rejection is practically zero.
"We are terribly excited by these results," he said.
"She is enjoying a normal life, which for us clinicians is the most beautiful gift."
Today Ms Castillo is living an active, normal life, and once again able to look after her children Johan, 15, and Isabella, four. She can walk up two flights of stairs without getting breathless.
I was a sick woman, now I will be able to live a normal life
Claudia Castillo
Professor Martin Birchall, professor of surgery at the University of Bristol who helped grow the cells for the transplant, said: "This will represent a huge step change in surgery.
"Surgeons can now start to see and understand the potential for adult stem cells and tissue engineering to radically improve their ability to treat patients with serious diseases."
He said that in 20 years time, virtually any transplant organ could be made in this way.
US scientists have already successfully implanted bladder patches grown in the laboratory from patients' own cells into people with bladder disease.
The European research team, which also includes experts from the University of Padua and the Polytechnic of Milan in Italy, is applying for funding to do windpipe and voice box transplants in cancer patients.
Clinical trials could begin five years from now, they said.
Between 50,000 and 60,000 people are diagnosed with cancer of the larynx each year in Europe, and scientists say about half them may be suitable candidates for tissue engineering transplants.